7.1 Convexity spaces and S-convexity
Many variations of Tverberg’s theorem appear if we change the underlying space we are using. For example, consider the following integer version of Tverberg’s theorem.
Theorem 7.1 (Integer Tverberg). Given r, d positive integers, there is an integer L = L(r, d) such that for any set of L points in Rd with integer coordinates there is a partition of the set intorpartsL1, . . . , Lrsuch that the intersection of their convex hulls contains a point with integer coordinates.
The exact values ofL(r, d) are not known, even forr= 2. The existence ofL(r, d) follows from results by Jamison [Jam81], as was noted by Eckhoff [Eck00]. The number of points needed is much larger than in the non-integer
case. For instance, we have the lower boundL(r, d)>(r−1)2d. To see this, taker−1 copies of each vertex of the hypercube [0,1]d. This lower bound may not always be optimal. For r = 2, Onn proved 542d+ 1 ≤ L(2, d) ≤ d(2d−1) + 3 [Onn91]. The cased= 3 remains interesting, withL(2,3)≤17 being the best upper bound [BB03]. The best general upper bound up to date is L(r, d)≤(r−1)d2d+ 1 [DLLHRS17b].
Problem 7.2. Determine the value L(2,3), or improve the current bounds 11≤L(2,3)≤17.
To state properly Tverberg’s theorem in abstract terms we only need two ingredients. First isCd, the family of all sets inRd that are considered convex, and the second is to be able to compute convex hull; i.e., an operator conv : 2Rd→ Cdwith a few properties. Thus, given a ground setY, a way to axiomatize convexity is to have an operator conv : 2Y →2Y which satisfies the following.
• conv(convA)) = convA for all A⊂Y,
• A⊂convA for all A⊂Y,
• A⊂B⊂Y implies convA⊂convB,
• For a countable sequenceA1⊂A2⊂. . .⊂Y we have that∪∞i=1conv(Ai) = conv (∪∞i=1Ai).
We say that the pair (Y,conv) is aconvexity space. A central questions of convexity spaces is the following.
Problem 7.3. Given a convexity space (Y,conv) and a positive integer t, determine the value of rt (if it exists) such that for anyX⊂Y of rt points there is a partition of X into t parts X1, . . . , Xt such that
t
\
j=1
convXj 6=∅.
The reason for the conflicting notation with our use of the variable r is that, in the context of convexity spaces, the numberrtabove is calledthet-th Radon number. A classic conjecture by Eckhoff was that for any convexity space with a finite r2 we have rt ≤ (t−1)(r2 −1) + 1. In other words, it asked if Tverberg’s theorem follows from Radon for purely combinatorial reasons. Hopes for this were dashed by an example, presented by Boris Bukh
in an unpublished preprint, that constructs a convexity space with r2 = 4 and rt≥3(t−1) + 2 [Buk10]. It remains open whetherrt can be bounded as a function that is linear in both r2 and t, which is enough for several applications.
General convexity spaces are outside the scope of this survey. The inter-ested reader should consult Eckhoff’s survey [Eck00] on the subject, which also discusses convexity spaces where Eckhoff’s conjecture is known to hold.
We focus on convexity spaces which are closely related to convexity inRd. An example is S-convexity, which generalizes the integer case. Given a set S ⊂Rd, we say that a set A⊂S is S-convex if A=S∩convA, where conv(·) denotes the usual convex hull in Rd. Given A ⊂ S, we define the S-convex hull convS(A) as the intersection of all S-convex setsB such that A⊂B.
A Tverberg-type theorem forSwould simply be the existence of a num-berTS(r, d) such that for any setXofTS(r, d) points ofS, there is a partition of them intor setsX1, . . . , Xr such that
r
\
j=1
convS(Xj)6=∅.
It turns out that the existence of such theorems relies on whether there is a Helly-type theorem forS-convexity [DLLHRS17b]. AnS-convexity Helly theorem simply says thatthere is a natural number h(S) such that, for any finite family of convex sets in Rd, if the intersection of any h(S) or fewer of them contains a point of S, then the intersection of the whole family contains a point ofS. The existence of Tverberg-type theorems is given by the following theorem.
Theorem 7.4. If convS has a Helly theorem with Helly numberh(S), then it has a Tverberg theorem. Moreover,TS(r, d)≤h(S)d(r−1) + 1 for allr.
This result implies the upper bound for integer Tverberg if we use Doignon’s theorem, which says that h(Zd) = 2d [Doi73, Sca77, Bel76]. If we have aquantitative Helly forS, i.e., a natural number hk(S) such that, for any finite family of convex sets in Rd, if the intersection of any hk(S) of them contains at least k points of S, then the intersection of the whole family contains at leastkpoints ofS, then we also obtain a similar Tverberg theorem, where now Tr
j=1convS(Xj) contains at least k points of S. The related Helly-type results on S-convexity are described in [ADLS17].
AlthoughS-convexity often gives worse bounds than the classic setting, it is interesting that in some cases the asymptotic behavior of variations of
Tverberg’s theorem remains the same. For example, we can naturally ask for a version of Tverberg with tolerance, Problem 4.11, makes sense in the integer lattice. We get the following result.
Theorem 7.5. Let r, t, d be positive integers, where r, d are fixed. Then, there is a number L(t) = rt+o(t) such that the following holds. Given a set X of L(t) points with integer coordinates in Rd, there is a partition of X into r sets X1, . . . , Xr such that for any for any C ⊂ X of cardinality at mostt, the convex hulls conv(X1\C), . . . ,conv(Xr\C) have a common point with integer coordinates.
For a proof, we only need to follow the methods of [GCRRP17] verba-tim. When they require the usage of a centerpoint, we simply need to use an integer point of depth 2−d, which exist as a consequence of Doignon’s theorem. It is interesting that for this version the linear-algebraic methods that use Sarkaria’s technique fail completely.
7.2 Tverberg-type theorem on families of sets.
It is possible to prove Tverberg-type results if we are dealing with families of subsets ofRd instead of just points. In this case, we have to replace the convex hulls by other operators, or require a conclusion stronger than the intersection of the convex hulls being non-empty. For example given a family F of d+ 1 hyperplanes in general position, we denote by ∆(F) the simplex whose faces are given by F. Then, we have the following Tverberg-type result by Karasev [Kar08, Kar11].
Theorem 7.6. Let r, d be positive integers such that r is a prime power.
Let F be a family of r(d+ 1) hyperplanes in general position in Rd. Then, there is a partition of F into r sets F1, . . . ,Fr of d+ 1 hyperplanes each such that
r
\
j=1
∆(Fj)6=∅
For this result there is also a corresponding discrete version of a center-point theorem for hyperplanes. Given a family of hyperplanesF and a point p∈Rd, we define the depth of pinF as the minimum number of members of F that a ray starting from p can hit. This was introduced in [RH99], and it was conjectured that every finite familyF of hyperplanes in general position in Rd has a pointp at depth greater than or equal to |F |/(d+ 1).
Theorem 7.6 implies that the answer is affirmative when |F |/(d+ 1) is a prime power.
Problem 7.7. Does Theorem 7.6 hold ifr is not a prime power?
Another family of variations appear if we have a family of convex sets which are large (i.e., they have large volume, large diameter, many lattice points, etc...) and we want to partition them so that the intersection of the convex hull of the parts is also a large convex set. We call thesequantitative versions of Tverberg’s theorem. Take for example the following Tverberg-type theorem for the diameter [Sob16a].
Theorem 7.8. Let r, dbe positive integers and ε >0 a real number. Then, there is a numberM =M(r, d, ε,diam)such that the following holds. Given a familyX ofM intervals of length 1in Rd, there is a partition ofX intor subfamilies X1, . . . , Xr such that the diameter of∩rj=1conv(∪Xj) is at least 1−ε. Moreover,M is linear inr.
The loss of diameter ε is necessary for this result. An equivalent state-ment can be proved for other functions [RS17], such as the volume. It is unclear if the lossεis still necessary in that setting.
Problem 7.9. Given r, d, determine if there is a number M(r, d,vol) such that the following holds. For any family C of M convex sets in Rd, each of volume at least one, there is a partition of C into r subfamilies C1, . . . ,Cr such that
vol
r
\
j=1
conv[ Cj
≥1.
Other interpretations of quantitative Tverberg appear in [DLLHRS17b, DLLHRS17a]. In those results, we are given a family of n “large” convex setsK1, . . . , Kn, and we seek a transversaly1 ∈K1, . . . , yn∈Knthat admits a (usual) Tverberg partition but where the intersection of the convex hulls of the parts is also “large”.
We can also significantly change the convexity in the conclusion of Tver-berg’s theorem. As mentioned in the introduction, if a subsetX of Rd with
|X|= (r−1)(d+ 1) + 1 is in sufficiently general position, then for a parti-tion of X into r setsX =X1∪. . .∪Xr 1≤ |Xj| ≤d+ 1 for every j, then Tr
j=1affXj is a single point.
Tverberg’s theorem simply says that we can always find a partition such that, if {p} = Tr
j=1affXj, the coefficients of the affine combination of Xj
that givep are non-negative. It turns out that sometimes we can prescribe some of those coefficients to be negative [BS17].
Theorem 7.10. Let X be a subset of Rd of (r −1)(d+ 1) + 1 points in sufficiently strong general position. Let M ⊂X be a set of points such that conv(M)∩conv(X\M) =∅. Then, there is a partition of X into r parts X1, . . . , Xrsuch that in the affine combinations that witnessTr
j=1affXj 6=∅, either
• all the coefficients forM are negative and all the coefficients forX\M are positive, or
• all the coefficients forM are positive and all the coefficients forX\M are negative.
Corollary 7.11. Assume that, under the conditions of the previous theorem,
|M|< r. Then in the affine combinations that witness Tr
j=1affXj 6=∅, all the coefficients for M are negative and all the coefficients for X \M are positive.
However, if we requirekcoefficients be negative, but we do not prescribe whichk points will carry the negative coefficients, the values ofkfor which this is possible is an open problem.
Problem 7.12. Find all triples of integers d, r, k for which the following holds. Given a subsetX ofRdof(r−1)(d+1)+1point in sufficiently general position, there is a partition ofX into r parts X1, . . . , Xr such that among the affine combinations that witness Tr
j=1affXj 6=∅, exactly k coefficients are negative.
8 Acknowledgments
This work was partly supported by the National Science Foundation un-der Grant No. DMS-1440140 while the first author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2017 semester. The first author was also supported by Hungarian Na-tional Research, Development and Innovation Office Grants no K111827 and K116769. The authors would like to thank Fr´ed´eric Meunier, Uli Wagner, G¨unter M. Ziegler, and an anonymous referee for their careful revision and helpful comments.
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